An example of pricing in competitive supply chain using association rule mining algorithm based on interval concept lattice

Abstract

The substitutability between products or the intensity of market competition is the key parameter affecting the supplier’s pricing decision. However, the parameter cannot be accurately measured in real life. This paper provides a method based on prior information to solve this issue. First, compared to classical concept lattice theory, the interval concept lattice theory can deal with uncertain information more accurately. It is used to extract the objects within the interval parameters [α, β], and then interval concepts and lattice structure are built. Second, based on the interval concepts and lattice structure, the association rule mining algorithm is designed to further extract the association rules under different interval parameters. Third, to obtain the effective association degree between two objects, the rule optimization algorithm is put forward by comparing the update of rules. Finally, the association degree can indirectly reflect the substitutability between products. Then the price of a new product can be determined. Our paper provides some implication on pricing for suppliers in competitive supply chain.

Keywords

Pricing decision formal context interval concept lattice structure optimization and mining of association rule

1 Introduction

In the competitive retail environment, pricing is a matter of supplier’s profitability. There is an enormous amount of literature focusing on pricing issues in supply chain. For example, Peng et al. [7] design the pricing strategies in a two-stage supply chain with two competitive manufacturers and one retailer. Li et al. [1] discuss the pricing strategies in both centralized and decentralized cases. They investigate the existence of the dual-channel supply chain. Li and Li [12] analyze the pricing policies of two players in a dual-channel supply chain, considering the value-added services to the product and fairness concerns of the retailer. The supplier’s optimal pricing strategies are investigated along with not only supply chain structure but also contribution (i.e., innovation and advertising) to the product demand. Either way, the optimal pricing is inseparable from the intensity of market competition (or the substitutability between products). Considering the immeasurability of the market intensity or the products substitutability, this paper designs the association rule mining algorithm based on interval concept lattice, in which the substitution degree between products can be indirectly reflected on the basis of the prior information [6]. Thus it provides a feasible method for pricing.

Association rule mining algorithm is a general method to acquire the correlation between data items in a database. The classical association rule mining algorithms mainly include Apriori algorithm, Dynamic Itemset Counting algorithm [16], Sampling algorithm [5] and Fast algorithm [15]. These association rules obtained by prior information are applied in a wide range of fields, such as business data analysis, e-commerce and so on. Concretely, if observing there is a strong correlation between two commodities from this algorithm, we would like to couple two commodities that can be purchased together to increase the sales of goods. Hu and Wang [19] present an association rule mining algorithm based on the concept lattice. It can certify all frequent itemsets and association rules with only one scan of databases. The innovation of our paper lies in mining association rule based on the interval concept lattice theory. Compared to the classical concept lattice [10 , 18], the interval concept lattice is a hierarchical structure of concept based on the interval parameters [α, β]. Each node in lattice structure is a formal concept, which reflects the unity of extension and intension of concept. In addition, the extensions of node in interval concept lattice are composed by two parts, upper extension and lower extension. The upper extension (or lower extension) contains objects covered by a certain-proportion set of attributes. It means to deal with uncertain information more flexibly and to express the uncertainty of items more accurately [2]. The relationship of each node implies the connection in generalization and specialization of concept. Therefore, interval concept lattice may serve as the basic data structure to mine more targeted association rules [17].

In reality, users generally care about objects covered by attributes under a certain amount or proportion [4 , 20], and thus mine more targeted association rules. Because the construction [3, 11], maintenance, combination and rules extraction of the interval concept lattice are all based on the given interval parameters [α, β], the values of interval parameters are very important. The current study shows that when α is roughly middle value, the lattice structure is the most stable. The precision and efficiency of extracted association rules are considerable. In our paper, first the interval concept lattice is built according to the formal context with decision attributes. The updating algorithm of lattice structure is designed with the equal-step change of interval parameters. Second, based on these concept lattice structures under different interval parameters, the association rules are successively obtained and the precisions of association rules are further calculated. Third, the optimal association degree is obtained by the rule optimization algorithm. Finally, our paper gives an example to illustrate the application on pricing in competitive supply chain.

2 Preliminary knowledge

2.1 Interval concept lattice theory

Definition 1. Based on a formal context (U, A, R) where U = {x₁, x₂, . . . , x_n} expresses an object set, A = {a₁, a₂, . . . , a_m} expresses an attribute set and R shows the binary relationship between U and A, the ternary sequence pair C (M^α, M^β, Y) is called interval concept. Assume that the interval parameters [α, β] satisfy the condition 0 ⩽ α < β ⩽ 1.

α - upper extension M^α: $M^{α} = {x | x \in M, | f (x) \cap Y | / | Y | ⩾ α, 0 ⩽ α ⩽ 1} .$

β - lower extension M^β: $M^{β} = {x | x \in M, | f (x) \cap Y | / | Y | ⩾ β, 0 ⩽ α < β ⩽ 1} .$

Among them, Y is the intension of the concept. |Y| is the number of elements contained by set Y, namely cardinal number. M^α expresses the objects covered by at least α × |Y| attributes from Y.

Definition 2. Using $L_{α}^{β} (U, A, R)$ to represent all interval concepts within [α, β], we record that $(M_{1}^{α}, M_{1}^{β}, Y_{1}) ⩽ (M_{2}^{α}, M_{2}^{β}, Y_{2}) \Leftrightarrow Y_{1} \supseteq Y_{2}$ where “⩽” is the partial ordering relation. If all concepts in $L_{α}^{β} (U, A, R)$ meet the partial ordering relation, call $L_{α}^{β} (U, A, R)$ is the interval concept lattice on the formal context (U, A, R).

In particular, when β = 1 (namely |f (x) ∩ Y|/|Y| = 1) and α = 0 (namely |f (x) ∩ Y|/|Y| = 0), the interval concept lattice $L_{α}^{β} (U, A, R)$ degenerates into a classical concept lattice L (U, A, R).

2.2 Association rule based on interval concept lattice

Association rule A ⇒ B is generated by the node binary group (C₁, C₂) where $C_{1} = C_{1} (M_{1}^{α}, M_{1}^{β}, A)$ , $C_{2} = C_{2} (M_{2}^{α}, M_{2}^{β}, A \cup B)$ and C₁ ⩾ C₂. Based on this node binary group, we can express the support θ of rule A ⇒ B as |Extension (C₂) |/|U| and the confidence φ of rule A ⇒ B as |Extension (C₂) |/|Extension (C₁) |.

Definition 3. If the node binary group (C₁, C₂) meets:

(i) |Extension (C₂) |/|U| ⩾ θ;

(ii) Extension (C₂) ⊆ Extension (C₁) and |Extension (C₂) |/|Extension (C₁) | ⩾ φ

The node binary group is called (θ, φ)-candidate binary group.

An interval concept contains two extensions, α- upper and β-lower extension. Based on it, α-upper and β-lower association rules can be extracted, respectively. The supports and confidences of rules can be calculated by the following.

α-upper association rules A ⇒ B: $Supp (A \Rightarrow B) = | M_{2}^{α} | / | U | and$ $Conf (A \Rightarrow B) = | M_{2}^{α} | / | M_{1}^{α} | .$

β-lower association rules A ⇒ B: $Supp (A \Rightarrow B) = | M_{2}^{β} | / | U | and$ $Conf (A \Rightarrow B) = | M_{2}^{β} | / | M_{1}^{β} | .$

Theorem 1. Association rule A ⇒ B can be call ed strong association rule, if it satisfies the conditions as follows.

A ∪ B is a frequent item set and Supp (A ⇒ B) ⩾θ;

Conf (A ⇒ B) ⩾ φ, namely |P (A ∪ B) |/|P (A) | ⩾φ.

Theorem 2. If C ⊂ B, rule A ⇒ C can be derived from rule A ⇒ B. The number of generated association rules can be effectively reduced.

From the perspective of interval concept lattice, association rules can be described by the implication relation between intensions, and indirectly reflect the approximate inclusion relation between extensions. Because the interval concept lattice is the unity of extension and intension with a certain proportion and reveals the inclusion relation between concepts, it is suitable to be the basic data structure of association rule mining.

2.3 Pricing strategy in competitive supply chain

We assume the demand function of product a is $Q_{a} = ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i} = ɛ - P_{a} + \sum_{i = 1}^{n} γ_{i} P_{i}$ where ɛ represents the market basis and γ_i reflects the association between product a and i . When γ_i > 0, an increase in the price P_i of product i leads to an increase in the order quantity Q_a of product a . We can regard that the relationship between product a and i is competitive or substitutable. If γ_i < 0, an increase in P_i leads to a decrease in Q_a, which implies the relationship between product a and i is complementary. There are two channel structure strategies designed:

(i) The supplier of product a sells directly to consumers and the profit function of supplier is $Π_{a} = Q_{a} P_{a} = (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + γ_{i} P_{i}) P_{a} .$ (1)

(ii) The supplier sells product a through a retailer and the profit functions of supplier and retailer are $Π_{a}^{S} = Q_{a} W_{a} = (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i}) W_{a},$ (2)

$\begin{matrix} and Π_{a}^{r} = Q_{a} (P_{a} - W_{a}) = \\ (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i}) (P_{a} - W_{a}), \end{matrix}$ (3) respectively.

In case (i), we can solve the following optimization problem defined by Equation (4) to obtain the optimal price. $\begin{matrix} P_{a}^{*} = {arg}_{P_{a}} max Π_{a} (P_{a} | P_{i}) \\ = (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i}) P_{a} \end{matrix} .$ (4)

In case (ii), we assume the optimization problem can be presented in the following two-stage programming. $\begin{matrix} max_{W_{a}} Π_{a}^{s} (W_{2}) = (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i}) W_{a} \\ ↓ \\ P_{a}^{*} = {arg}_{P_{a}} max Π_{a}^{r} (P_{a} | W_{a}) \\ = (ɛ - P_{a} + γ_{1} P_{1} + γ_{2} P_{2} + . . . γ_{i} P_{i}) (P_{a} - W_{a}) . \end{matrix}$ (5)

Generally, the parameter ɛ can be regarded as 1. In addition, the association coefficient γ_i plays important role on the pricing strategy.

3 Association rule mining algorithm based on interval concept lattice

A key link is to build interval concept lattice structure before mining association rules. Interval concept lattice structure can change with the respect to the interval parameters [α, β]. Therefore, we can obtain a rule base containing all association rules under different interval parameters. In the following subsections, first, the updating algorithm of lattice structure is briefly introduced. And then the association rules can be extracted based on concept lattice with different interval parameters. Further our study explores the optimal association rules depending on the availability, supports and confidences of rules for the purpose of application.

3.1 Interval concept lattice updating algorithm with the change of parameters

When the interval parameters change, it is cumbersome to reconstruct the lattice structure on the basis of changed parameters. Thus, we design the updating algorithm for greater efficiency. The updating of interval concept lattice is mainly divided into two parts. One is caused by the change of formal context, and the other is caused by the change of interval parameters. In this section we focus on updating the interval concept lattice along with the adjustment of the parameters. For instance, the parameters change into [α₁, β₁] from [α₀, β₀], and then there are four kinds of cases: (1) α₁ > α₀; (2) α₁ < α₀; (3)β₁ > β₀; (4)β₁ < β₀.

The changes in cases (1) and (2) need to update the upper extension of the interval concept, M^{α
₀} → M^{α
₁}; the other changes need to update the lower extension of the interval concept, M^{β
₀} → M^{β
₁}. Therefore, four functions are given as follows.

(1) Function: CL1 (C, α₀, α₁) //C is any node in concept lattice, and α₁ > α₀

{Ma = {Ø}

For each x in M^{α
₀} of C:

If $\frac{| f (x) \cap Y |}{| Y |} ⩾ α_{1}$ then Ma = Ma ∪ x

M^{α
₁} = Ma}

(2) Function: CL2 (C, α0, α1) //C is any node in concept lattice; and α1 < α₀.

{Ma = M^{α
₀}

For the upper extension Maf of any father-node CF in C:

{ Make maf1 = Maf - M^{α
₀}

For ∀x ∈ maf1:

If $\frac{| f (x) \cap Y |}{| Y |} \geq α_{1}$ then Ma = Ma ∪ x//Y is the intension set of C

M^{α
₁} = Ma}

(3) Function: CL3(C, β₀, β₁) //C is any node in concept lattice; and β₁ > β₀.

{ Mb = {Ø}

For each x in M^{β
₀} of C

If $\frac{| f (x) \cap Y |}{| Y |} \geq β_{1}$ then Mb = Mb ∪ x

M^{β
₁} = Mb}

(4) Function: CL4 (C, β₀, β₁) // C is any node in concept lattice; and β₁ < β₀.

{Mb = M^{β
₀}.

For the upper extension Mbf of any father-node CF in C:

{ Make mbf1 = Mbf - M^{β
₀}

For ∀x ∈ mbf1:

If $\frac{| f (x) \cap Y |}{| Y |} \geq β_{1}$ then Mb = Mb ∪ x} // Y is the intension set of C

M^{β
₁} = Mb}

Based on the four functions, when the interval parameters change, we use the method of breadth-first to visit and judge each node from the root node in lattice structure. According to the above cases, we update and adjust the nodes, including deleting the redundancy concepts and empty concepts from the lattice structure, and adjusting the father-son relationship.

Algorithm 1: Updating algorithm of interval concept lattice based on parameters

Input: Formal context (U, A, R), initial lattice structure $L_{α_{0}}^{β_{0}} (U, A, R)$ , interval parameters (α₁, β₁)

Output: $L_{α_{1}}^{β_{1}} (U, A, R)$

Step 1, C₁ = (M^α, M^β, Y) is the root node of $L_{α_{0}}^{β_{0}} (U, A, R)$ . If Y =∅, C₁ does not change; if Y≠ ∅ Y and α₁ > α₀, we call function: CL1 (C, α₀, α₁), else call function: CL2 (C, α₀, α₁), and then update M^{α
₀} to M^{α
₁}. Similarly, if β₁ > β₀, we call function: CL3 (C, β₀, β₁), else call function: CL4 (C, β₀, β₁), and then update M^{β
₀} to M^{β
₁} and C₁ to (M^{α
₁}, M^{β
₁}, Y).

Step 2, Visit each children-nodes C_i in C₁.

Step 3, Assume $C_{1} = (M_{i}^{α}, M_{i}^{β} Y_{i})$ . If α₁ > α₀, we call function: CL1(C, α₀, α₁), else call function: CL2(C, α₀, α₁), and then update $M_{i}^{α_{0}}$ to $M_{i}^{α_{1}}$ ; if $M_{i}^{α_{1}} = φ$ , we delete node C_i, else continue updating the lower extension. If β₁ > β₀, we call function: CL3(C, β₀, β₁), else call function: CL4(C, β₀, β₁), and then update $M_{i}^{β_{0}}$ to $M_{i}^{β_{1}}$ , and C_i is totally updated to ( $M_{i}^{α_{1}}, M_{i}^{β_{1}}, Y_{i}$ ).

Step 4, For each father-node $C_{i} (M_{i}^{α_{1}}, M_{i}^{β_{1}}, Y_{i}) \to parent$ in C_i, if $M_{i}^{’ β_{1}} = M_{i}^{β_{1}}$ and $C_{i} \to parent = C_{i}^{'} \to parent$ , we delete C_i’.

Step 5, For each children-node in C_i, if , turn to “Step 3” until visiting the final node in $L_{α_{0}}^{β_{0}} (U, A, R)$ .

Step 6, Output the new concept lattice structure: $L_{α_{1}}^{β_{1}} (U, A, R)$ .

Step 7, End.

Based on the original interval concept lattice, if changing interval parameters, the extension of the local nodes will change. The updating algorithm aims at traversing each node of the original structure, retaining or updating nodes extension, and then obtaining new concept lattice. Compared to the reconstruction of lattice structure, the updating algorithm is better than reconstruction in time complexity.

3.2 Association rule mining algorithm based on interval concept lattice

Based on a given interval concept lattice structure, α-upper and β-lower association rules can be extracted, respectively, due to the upper and lower extension of interval concept. Next, we will give an example of extracting α-upper association rules as the following. Similarly, the algorithm of extracting β-lower association rules can be designed.

Algorithm idea: The interval concept lattice is regarded as a kind of data structure. According to it, we can design the association rule extraction algorithm. After inputting every node (interval concept C (M^α, M^β, Y)) in lattice structure, for each child node C of the root node, we traverse all of its successor nodes and find the bottom node C^*. All condition attributes from the node are as the left one in a rule, and all decision attributes are as the right one in a rule. An association rule can be obtained.

Algorithm 2: α-upper association rules mining algorithm based on interval concept lattice

Input: Interval parameter α, lattice $L_{α}^{1} (U, A, R)$ , minimum support and confidence thresholds θ and φ.

Output: α-upper association rules.

Step 1, Traverse interval concept lattice $L_{α}^{1} (U, A, R)$ by breadth-first search algorithm and obtain α-upper frequent nodes set, α-Fcset. For any concept node C₁ = (M^α, M¹, Y) in lattice structure, if |M^α| ⩾ |U| × θ, there is α-Fcset=α-Fcset ∪ {C₁}. Finally upper frequent nodes set can be obtained after traversing all interval concepts.

Step 2, Generate all α-upper candidate binary groups. Assume the nodes set of candidate binary groups is PAIRS(C₁). It is formed with any node $C_{1} = (M_{1}^{α}, M_{1}^{1}, Y_{1})$ in α-upper frequent nodes set. For other node $C_{2} = (M_{2}^{α}, M_{2}^{1}, Y_{2})$ in α-Fcset, if C₂ > C₁ and $| M_{1}^{α} | / | M_{2}^{α} | ⩾ φ$ , we have PAIRS(C₁)=PAIRS (C₁)∪ {C₂}. Repeat above steps until finding all PAIRS in α-Fcset.

Step 3, Eliminate superfluous candidate binary groups. Arrange the nodes of α-Fcset in the descending order of the intension cardinal numbers. If C₁ > C₂, there is PAIRS(C₁)=PAIRS(C₁) – PAI RS(C₂). After eliminating superfluous of α-Fcset, α-upper candidate binary groups are obtained, and then α-upper frequent nodes set α-Fcset is recorded.

Step 4, Generate α-upper association rules: α - Rules set.

Step 5, End.

Based on this algorithm we can extract more refined uncertain association rules, and improve the reliability of the rules. In reality, users often need some association rules that are easy to understand and can better reflect the real situation.

3.3 Rule optimization algorithm based on interval concept lattice

Original rule base is usually built by experts who set the parameters according to their own experience. However, it could not describe the nonlinear relationship between input and output accurately. We can use the sample data to train the original system and obtain the corresponding rules under the given parameters. According to the relation between input and output, the optimization of rules will be realized, which can further enhance the performance of the rule base system. Here the algorithm will be introduced to find the relationship between interval parameters [α, β] and rules, and output the optimal association rules.

Algorithm idea: Through training formal context, we examine the optimal interval parameters and output the optimal rules on the basis of the given formal context. First, according to the number n of attributes in formal context, we divide parameter α by the equal step whose length λ is 1/n, and set α_i = i/n, where i = 1, 2, 3 … n. To facilitate the description of the algorithm, we set up the initial parameters values:α₀ = 1/n, β₀ = 1, and then build concept lattice and extract association rules. When the interval parameters change by the equal step, we can update the original lattice structure to obtain the new concept lattice structure and association rules. Finally we find the relationship between interval parameters and association rules. When a new formal context M’ is input, the optimal rules will be directly output based on the trained parameters.

Algorithm 3: Rule optimization algorithm based on the interval concept lattice

Input: the training sample: formal context M; the test sample: formal context M’; the minimum support and confidence thresholds θ and φ.

Output: The optimal rules based on M’.

Step 1, According to the formal context M, we obtain the number of attribute n, and set up the length of step λ = 1/n. Initialize the interval parameters α0 = 1/n, β0 = 1, and then $L_{α_{0}}^{β_{0}} (U, A, R)$ is built based on Algorithm 1.

Step 2, Put $L_{α_{0}}^{β_{0}} (U, A, R)$ , θ and φ as the input of Algorithm 2. Output the result of Algorithm 2 and record the number n₀ and confidence φ₀ of rules.

Step 3, Make α₁ = α₀ + λ. Based on Algorithm 1, we update the lattice structure, and then turn to Step 2 to record the numbers n₁ of association rules in which the confidence has changed.

Step 4, Set τ₀, If n₁/n₀ > τ₀, turn to Step 3; else record the interval parameter α_i.

Step 5, Put the formal context M and parameter α_i as the input of Algorithm 1. And then input the results of Algorithm 1 into Algorithm 2 to obtain the optimal rules.

Step 6, End.

For the rules optimization issue, we try to find the specific function relation between interval parameters and association rules, but there is complicated nonlinear relationship between them. The specific function relation is uncertain and hard to get. Therefore, it is feasible to establish a dynamic rules optimization model to deal with the issue. According to the given formal context, calculation times of the model are not more than the number of attributes n, and all of the concept nodes do not have to rebuild when updating the interval concept lattice. It greatly reduces the time complexity of the model. Although the calculation is tedious in extracting association rules of concept nodes, to some extent, it keeps the precision of the model. When the formal context is too complicated, the efficiency of the model may be reduced.

4 Example analysis

In reality, consumers’ purchasing behaviors indirectly reflect the competition intensity between products. As Table 1, the formal context shows the purchasing behaviors of ten objects, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {a, b, c, d} is a condition attributes set and B = {e} expresses a decision attribute set. For example, the second consumer buys products b and e, but not products a, c and d during shopping. For simplicity, we fix the value of parameter β is 1 and explore the association rules with the respect of parameter α.

Table 1
Formal context

Objects a b c d e

1 1 0 1 1 0

2 0 1 0 0 1

3 1 0 1 1 0

4 0 1 0 1 1

5 1 1 1 0 1

6 1 0 0 1 1

7 1 0 1 1 1

8 0 0 1 1 0

9 1 1 1 0 1

10 0 1 0 1 0

Objects	a	b	c	d	e
1	1	0	1	1	0
2	0	1	0	0	1
3	1	0	1	1	0
4	0	1	0	1	1
5	1	1	1	0	1
6	1	0	0	1	1
7	1	0	1	1	1
8	0	0	1	1	0
9	1	1	1	0	1
10	0	1	0	1	0

For the sake of comparison and illustration, we enumerate all the corresponding interval concept lattice structures and association rules according to the algorithms in the model.

Initialize the interval parameter α=1/5. Based on interval parameters [1/5,1], we obtain the concepts in Table 2 and then build the interval concept lattice structure as Fig. 1 showing. From Table 2, we find that the M^1/5(1/5-upper extension) of each concept contains objects whose number is greater than 5. In other words, we can deduce an association rule from almost every relationship between father-node and son-node. The results of mining association rule are showing as Table 7, where the confidence of each rule is 1. The parameter α is so low that the concept node contains too much invalid information. These association rules are almost ineffective. On the one hand, according to the formal context, we observe that six of the ten consumers purchase product a, while four of those six consumers purchase product e as well. It is intuitive to draw a conclusion that the consumer who buys product a has a two-thirds chance of buying product e. In contrast, the confidence of association rule a⇒e from Table 7 is unauthentic. On the other hand, when the parameter α changes into 2/5 from 1/5, for instance, the confidence of rule a b⇒e will change into 2/3 from 1. It implies that the extracted rule base is not stable. Although the case with α=1/5 makes the best use of all the information in formal context, most of rules are redundant. According to Algorithm 3, we should make α₁ = α₀ + λ.

Table 2

[1/5,1] interval concept nodes

Interval concept	Upper extension	Lower extension	Intension	Interval Concept	Upper extension	Lower extension	Intension
C1	{12345678910}	{12345678910}	Ø	C17	{12345679}	{5679}	{ae}
C2	{135679}	{135679}	{a}	C18	{24567910}	{2459}	{be}
C3	{245910}	{245910}	{b}	C19	{12345678910}	{579}	{ce}
C4	{135789}	{135789}	{c}	C20	{12345678910}	{467}	{de}
C5	{13467810}	{13467810}	{d}	C21	{1234567910}	{59}	{abe}
C6	{1234567910}	{59}	{ab}	C22	{123456789}	{579}	{ace}
C7	{1356789}	{13579}	{ac}	C23	{12345678910}	{67}	{ade}
C8	{1345678910}	{1367}	{ad}	C24	{12345678910}	{59}	{bce}
C9	{1234578910}	{59}	{bc}	C25	{12345678910}	{4}	{bde}
C10	{12345678910}	{410}	{bd}	C26	{12345678910}	{7}	{cde}
C11	{1345678910}	{1378}	{cd}	C27	{12345678910}	{59}	{abce}
C12	{12345678910}	{59}	{abc}	C28	{12345678910}	Ø	{abde}
C13	{12345678910}	Ø	{abd}	C29	{12345678910}	{7}	{acde}
C14	{1345678910}	{137}	{acd}	C30	{12345678910}	Ø	{bcde}
C15	{12345678910}	Ø	{bcd}	C0	{12345678910}	Ø	{abcde}
C16	{12345678910}	Ø	{abcd}

Fig. 1

, [2/5,1] or [3/5,1] interval concept lattice.

When the interval parameter α=2/5, we find the number of nodes (see Table 3) and the relationship between father-node and son-node (see Fig. 1) are not changing, but the confidences of some rules are changing. For example, from Table 7 the confidence of rule ab⇒e is changing into 2/3 from 1. According to the formal context, we observe that only two customers purchase products a and b, and both of them buy product e as well. Obviously, the confidence of rule ab⇒e is 1 that is closer to the reality. Moreover, the number of association rules whose confidence has changed is more than 15 × τ. Therefore, we should make α₂ = α₁ + λ.

Table 3

[2/5,1] interval concept nodes

Interval concept	Upper extension	Lower extension	Intension	Interval concept	Upper extension	Lower extension	Intension
C1	{12345678910}	{12345678910}	Ø	C17	{12345679}	{5679}	{ae}
C2	{135679}	{135679}	{a}	C18	{24567910}	{2459}	{be}
C3	{245910}	{245910}	{b}	C19	{12345678910}	{579}	{ce}
C4	{135789}	{135789}	{c}	C20	{12345678910}	{467}	{de}
C5	{13467810}	{13467810}	{d}	C21	{245679}	{59}	{abe}
C6	{1234567910}	{59}	{ab}	C22	{135679}	{579}	{ace}
C7	{1356789}	{13579}	{ac}	C23	{1345679}	{67}	{ade}
C8	{1345678910}	{1367}	{ad}	C24	{24579}	{59}	{bce}
C9	{1234578910}	{59}	{bc}	C25	{24567910}	{4}	{bde}
C10	{12345678910}	{410}	{bd}	C26	{13456789}	{7}	{cde}
C11	{1345678910}	{1378}	{cd}	C27	{12345679}	{59}	{abce}
C12	{13579}	{59}	{abc}	C28	{1234567910}	Ø	{abde}
C13	{134567910}	Ø	{abd}	C29	{13456789}	{7}	{acde}
C14	{1356789}	{137}	{acd}	C30	{12345678910}	Ø	{bcde}
C15	{134578910}	Ø	{bcd}	C0	{12345678910}	Ø	{abcde}
C16	{1345678910}	Ø	{abcd}

When the interval parameter α=3/5, we find the extension of each node in Table 4 becomes more accurate than that under the case where α = 1/5 or α = 2/5. In addition, the confidences of rules have great changes, that is, almost every rule’s confidence reduces. On the one hand, the relatively low parameter α can lead to more invalid information being included. We can observe the confidences for most of rules in the case where α = 1/5 or α = 2/5, are close to 1. Obviously, the accuracies for these rules will be questioned. On the other hand, the relatively high α restricts some objects from entering the extension of node. However, to some degree, it ensures the accuracy of the rules. Therefore, the criterion for selecting the optimal α is not only to provide users with a lot of choices but also to make rules high-precision. From the case where α = 4/5 or α = 5/5, we conclude that the optimal value of parameter α is 3/5 (see, for example, [8] Li et al., 2016).

Table 4

[3/5,1] interval concept nodes

Interval concept	Upper extension	Lower extension	Intension	Interval concept	Upper extension	Lower extension	Intension
C1	{12345678910}	{12345678910}	Ø	C17	{5679}	{5679}	{ae}
C2	{135679}	{135679}	{a}	C18	{2459}	{2459}	{be}
C3	{245910}	{245910}	{b}	C19	{579}	{579}	{ce}
C4	{135789}	{135789}	{c}	C20	{467}	{467}	{de}
C5	{13467810}	{13467810}	{d}	C21	{245679}	{59}	{abe}
C6	{59}	{59}	{ab}	C22	{135679}	{579}	{ace}
C7	{13579}	{13579}	{ac}	C23	{1345679}	{67}	{ade}
C8	{1367}	{1367}	{ad}	C24	{24579}	{59}	{bce}
C9	{59}	{59}	{bc}	C25	{24567910}	{4}	{bde}
C10	{410}	{410}	{bd}	C26	{13456789}	{7}	{cde}
C11	{1378}	{1378}	{cd}	C27	{59}	{59}	{abce}
C12	{13579}	{59}	{abc}	C28	{45679}	Ø	{abde}
C13	{134567910}	Ø	{abd}	C29	{135679}	{7}	{acde}
C14	{1356789}	{137}	{acd}	C30	{4579}	Ø	{bcde}
C15	{134578910}	Ø	{bcd}	C0	{1345679}	Ø	{abcde}
C16	{1345678910}	Ø	{abcd}

In order to further clarify the optimal parameter obtained by the algorithm, we analysis the case of α = 4/5 (see Table 5) or α = 5/5 (similar to the case of α = 4/5 and omitted due to space limitation). According to Table 7, the number of association rules falling sharply leads to a narrow users’ choice space, which is not good for mining potential and useful information. Given the previous statement, the value of α is the best in the middle range to achieve the optimal rules. In this example, the best range is [3/5 4/5). Further adjustment in parameters and rules can be realized according to the specific needs of users. As far as possible improve the precision of the rules when the number of association rules is close to the users’ satisfaction.

Table 5

[4/5,1] interval concept nodes

Interval concept	Upper extension	Lower extension	Intension	Interval concept	Upper extension	Lower extension	Intension
C1	{12345678910}	{12345678910}	Ø	C17	{5679}	{5679}	{ae}
C2	{135679}	{135679}	{a}	C18	{2459}	{2459}	{be}
C3	{245910}	{245910}	{b}	C19	{579}	{579}	{ce}
C4	{135789}	{135789}	{c}	C20	{467}	{467}	{de}
C5	{13467810}	{13467810}	{d}	C21	{59}	{59}	{abe}
C6	{59}	{59}	{ab}	C22	{579}	{579}	{ace}
C7	{13579}	{13579}	{ac}	C23	{67}	{67}	{ade}
C8	{1367}	{1367}	{ad}	C24	{59}	{59}	{bce}
C9	{59}	{59}	{bc}	C25	{4}	{4}	{bde}
C10	{410}	{410}	{bd}	C26	{7}	{7}	{cde}
C11	{1378}	{1378}	{cd}	C27	{59}	{59}	{abce}
C12	{59}	{59}	{abc}	C28	Ø	Ø	{abde}
C13	Ø	Ø	{abd}	C29	{7}	{7}	{acde}
C14	{137}	{137}	{acd}	C30	Ø	Ø	{bcde}
C15	Ø	Ø	{bcd}	C0	{579}	Ø	{abcde}
C16	Ø	Ø	{abcd}

From Table 7 rwe find that the optimal rules include a⇒e rb⇒e rc⇒e and d⇒e rin which their confidences are 2/3 r4/5 r1/2 and 3/7 rrespectively. In other words ra consumer who has purchased product a will have a two-thirds chance of buying product e. On the contrary rhe will have a one-third chance of not buying product e rwhich implies that the substitutability between products a and e is one-third. In general rwhen two products are completely substitutable rthe customer just chooses to buy one of them. Therefore rbased on Section 2.3 and the priori information rwe can give the pricing policy depending on the competitive products a b c d respectively.

Table 7

Association rules with change of interval parameters

Interval parameters	Association rules	confidence	Association rules	Accuracy	Association rules	confidence
	A⇒e	1	ad⇒e	1	acd⇒e	1
	b⇒e	1	bc⇒e	1	bcd⇒e	1
[1/5,1]	c⇒e	1	bd⇒e	1	abcd⇒e	1
	d⇒e	1	cd⇒e	1
	ab⇒e	1	abc⇒e	1
	ac⇒e	1	abd⇒e	1
	a⇒e	1	ad⇒e	7/9	acd⇒e	1
	b⇒e	1	bc⇒e	5/9	bcd⇒e	1
[2/5,1]	c⇒e	1	bd⇒e	7/10	abcd⇒e	1
	d⇒e	1	cd⇒e	8/9
	ab⇒e	2/3	abc⇒e	1
	ac⇒e	6/7	abd⇒e	1
	a⇒e	2/3	ad⇒e	1	acd⇒e	6/7
	b⇒e	4/5	bc⇒e	1	bcd⇒e	1/2
[3/5,1]	c⇒e	1/2	bd⇒e	1	abcd⇒e	7/9
	d⇒e	3/7	cd⇒e	1
	ab⇒e	1	abc⇒e	2/5
	ac⇒e	1	abd⇒e	5/8
	a⇒e	2/3	ad⇒e	1/2	acd⇒e	1/3
	b⇒e	4/5	bc⇒e	1	bcd⇒e	/
[4/5,1]	c⇒e	1/2	bd⇒e	1/3	abcd⇒e	/
	d⇒e	3/7	cd⇒e	1/4
	ab⇒e	1	abc⇒e	1
	ac⇒e	3/5	abd⇒e	/
	a⇒e	2/3	ad⇒e	1/2	acd⇒e	1/3
	b⇒e	4/5	bc⇒e	1	bcd⇒e	/
[5/5,1]	c⇒e	1/2	bd⇒e	1/3	abcd⇒e	/
	d⇒e	3/7	cd⇒e	1/4
	ab⇒e	1	abc⇒e	1
	ac⇒e	3/5	abd⇒e	/

When faced with a competitive product i (i = a, b, c or d), the pricing strategies of product e can be shown as follows.

Case (i): Both suppliers sell directly to consumers.

The profit functions of both suppliers are Π_i = (1 - P_i + γP_e) P_i and Π_e = (1 - P_e + γP_i) P_e respectively. According to Equation (4) we can obtain the optimal price of product e as follows. $P_{e}^{*} = \frac{1 + γ P_{i}^{*}}{2} = \frac{1}{2 - γ} .$

Simultaneously, the optimal profit for product e is $Π_{e}^{*} = \frac{1}{{(2 - γ)}^{2}}$ . It is easy to see that the profitability of product e is mainly influenced by the market intensity or the substitutability between products

Case (ii): One of both suppliers sells directly to consumers while the other sells through a retailer.

Due to symmetry, we assume product i is sold directly and product e is sold by a retailer, and the profit functions of firms are Π_i = (1 - P_i + γP_e) P_i and Π_e = (1 - P_e + γP_i) W_e, respectively. On the basis of Equation (5) we can give the optimal price of product e as follows.

$P_{e}^{*} = \frac{2 P_{i}^{*} - 1}{γ} = \frac{3 - γ^{2}}{(2 - γ) (2 - γ^{2})}$ . And the optimal profit of supplier which sells product e is $Π_{e}^{*} = \frac{2 + γ}{4 (2 - γ) (2 - γ^{2})}$ .

Case (iii): One of both suppliers sells through a retailer, while the other sells directly to consumers.

Similarly, we assume product e is sold directly and product i is sold by a retailer and obtain the optimal price and profit of product e as follows. $\begin{matrix} P_{e}^{*} = \frac{1 + γ P_{i}^{*}}{2} = \frac{4 + γ - 2 γ^{2}}{2 (2 - γ) (2 - γ^{2})} and \\ Π_{e}^{*} = {[\frac{4 + γ - 2 γ^{2}}{2 (2 - γ) (2 - γ^{2})}]}^{2} . \end{matrix}$

Case (iv): Both suppliers sell through retailers.

The profit functions of suppliers are Π_i = (1 - P_i + γP_e) W_i and Π_e = (1 - P_e + γP_i) W_e, respectively. According to Equation (5) we can obtain the optimal price of product e as follows. $\begin{matrix} P_{e}^{*} = \frac{2 (3 - γ^{2}) + γ (4 - γ - 2 γ^{2}) P_{i}^{*}}{2 (4 - γ - 2 γ^{2})} \\ = \frac{2 (3 - γ^{2})}{(2 - γ) (4 - γ - 2 γ^{2})} \end{matrix} .$

And the optimal profit of firm which sells product e is $Π_{e}^{*} = \frac{(2 + γ) (2 - γ^{2})}{(2 - γ) {(4 - γ - 2 γ^{2})}^{2}}$ .

Considering the confidences of association rules in Table 7, we can calculate the price and profit of product e when faced with different competitors. The results are shown as follows. Specifically, we design the pricing policy under different competitors and selling contexts. From Table 8, we can give the following management implications.

Table 8

Price and profit of product e

Competitor	case	(i)	(ii)	(iii)	(iv)
a	price	0.7500	1.2321	0.9107	1.5682
	profit	0.5625	0.3214	0.8294	0.5207
b	price	0.8333	1.4461	1.0784	2.0486
	profit	0.6944	0.4289	1.1630	0.8608
c	price	0.6667	1.0476	0.7619	1.2222
	profit	0.4444	0.2381	0.5805	0.3241
d	price	0.6364	0.9867	0.7114	1.1187
	profit	0.4050	0.2127	0.5061	0.2734

From the profitability’s perspective, no matter how the substitutability between product i (i = a, b, c or d) and e is, the supplier of product e who sells directly to consumers will be most profitable when her competitor sells through a retailer (i.e., 0.8294 > 0.5625 > 0.5207 > 0.3214). In addition, the retail price of product e is highest when both supplier i and supplier e choose to sell through respective retailer. This is because the middleman raises the retail price. On the contrary, the retail price of product e is lowest when both supplier i and supplier e sell directly to consumers.

Generally, the products sold through retailers are priced higher than those sold directly to consumers. For instance, when faced with the competitor a, the retail price of product e in case (iv) is more than that in case (i), namely, 0.7500 < 1.5682. However, given the profitability, we observe that higher prices do not necessarily lead to higher profits. For instance, when product e is faced with product a (c or d), we find the profitability from product e sold by a retailer is lower than that sold directly (i.e., 0.5207 < 0.5625, 0.3241 < 0.4444 or 0.2734 < 0.40 50). Significantly, when faced with the competitor b, the profit of product e sold by a retailer is higher than that sold directly (i.e., 0. 8608 > 0.6944). The reason behind this is that the relatively high substitutability (0.8) between product b and e results in an increase in order quantity of product e.

According to the data in the Table 8, we find the pricing of product e is influenced by the substitutability depending on who its competitor is, selling strategies (direct or indirect) of itself and its competitor, and its own profitability. Specifically, the lowest price for product e occurs when the substitutability is 3/7 and both suppliers sell directly. At this moment, the supplier’s profitability of product e is not the lowest. By contrast, the highest price for product e occurs when the substitutability is 4/5 (the highest of four substitution levels) and both suppliers sell through retailers. In reality, suppliers usually prefer to focus on their profitability. When faced with product b which is sold by a retailer, the supplier of product e being sold directly will make maximum profit 1.1630. Besides, the supplier of product e sold through a retailer will make minimum profit 0.2127 facing a competitor who sells product b directly to consumers. Obviously, the direct sales mode has an advantage in profitability relative to the indirect sales mode.

5 Conclusion

This paper focuses on the pricing issue in competitive supply chain by using association rule mining algorithm based on interval concept lattice. On the basis of given decision formal context, the concept lattices are built under different parameters, and then a lot of association rules are mined. It is easy to find that the change of interval parameters will directly affect the interval concept lattice structure. And then the number and precision of extracted association rules will change with the respect of the interval parameters. Through analyzing the training formal context, rules optimization model is built and the rough influence law between parameter α and association rules is present in detail. It helps provide the optimal interval parameter and further extract the effective association rules. We find that when the parameter α is in [3/5, 4/5), the optimal association rules can be obtained. What is more, according to these optimal rules, some association degree between objects will be obtained, which is the key to solve the pricing issue. In particular, when the association degree (or be called the substitutability) between both products is relatively high and they are sold through retailers, the retail price is the highest but the profitability is not the lowest.

Despite the importance of the association degree between objects on pricing, our study also has a few limitations. In our paper, we assume the demand of product is a linear function. However, in real life we can regard the demand as stochastic or multi-mode 9]. Accordingly, we will continue the study of pricing issue under the uncertain demand.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 71971113).

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An example of pricing in competitive supply chain using association rule mining algorithm based on interval concept lattice

Abstract

Keywords

1 Introduction

2 Preliminary knowledge

2.1 Interval concept lattice theory

2.2 Association rule based on interval concept lattice

2.3 Pricing strategy in competitive supply chain

3.1 Interval concept lattice updating algorithm with the change of parameters

3.2 Association rule mining algorithm based on interval concept lattice

3.3 Rule optimization algorithm based on interval concept lattice

4 Example analysis

Table 1 Formal context Objects a b c d e 1 1 0 1 1 0 2 0 1 0 0 1 3 1 0 1 1 0 4 0 1 0 1 1 5 1 1 1 0 1 6 1 0 0 1 1 7 1 0 1 1 1 8 0 0 1 1 0 9 1 1 1 0 1 10 0 1 0 1 0

Footnotes

Acknowledgments

References

Table 1
Formal context

Objects a b c d e

1 1 0 1 1 0

2 0 1 0 0 1

3 1 0 1 1 0

4 0 1 0 1 1

5 1 1 1 0 1

6 1 0 0 1 1

7 1 0 1 1 1

8 0 0 1 1 0

9 1 1 1 0 1

10 0 1 0 1 0